Finite Group Databasebeta

← #320657

Subgroups — #320657

Order 512.

889 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 889 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33230C211yesmaximal, characteristic, complemented
2#29139C212yesmaximal, complemented
3#28615C212yesmaximal, complemented
4#21690C211yesmaximal, characteristic, complemented
5#10686C211yesmaximal, characteristic, complemented
6#10677C211yesmaximal, characteristic, complemented
7#10307C211yesmaximal, characteristic, complemented
8#10304C211yesmaximal, characteristic
9#9838C212yesmaximal, complemented
10#9832C212yesmaximal, complemented
11#8263C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#493C2212yes—
14C2×#489—22no—
15C2×#486C2212yes—
16C2×#442—22no—
17#2868C2211yescharacteristic
18#2858C2211yescharacteristic
19C4×#127C2212yes—
20#2054—24no—
21#2023—24no—
22#1990C2212yes—
23#1972—22no—
24#1959—22no—
25#1957C2212yes—
26#1957C2211yescharacteristic
27#1957C2212yes—
28#1957C2211yescharacteristic
29#1957C2212yes—
30#1957C2212yes—
31#1956C2211yescharacteristic
32#1956C2211yescharacteristic
33#1879—24no—
34#1844C2211yescharacteristic
35#1834C2212yes—
36#1834C2212yes—
37#1802—24no—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×D8—22no—
44C22×D8—24no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#127C2311yescharacteristic
49C2×#127C2312yes—
50C2×#124—21no—
51C2×#124—21no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55#493—24no—
56#493—24no—
57#493—42no—
58#489—48no—
59#486—24no—
60#486—42no—
61#442—48no—
62C2×#106—22no—
63C2×C4⋊C8D414yes—
64C2×#103—22no—
65C2×#103—22no—
66C2×#102—24no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#97D412yes—
74C2×#97D412yes—
75C2×C4×C8C2312yes—
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#390—22no—
80#390—22no—
81#390—22no—
82#390—22no—
83#390—22no—
84#384—22no—
85#384—21no—
86#384C2312yes—
87#384C2311yescharacteristic
88#384C2311yescharacteristic
89C43C2311yescharacteristic
90#340—24no—
91#340—24no—
92#340—24no—
93#336D414yes—
94#322—42no—
95C22×Q8—21no—
96C22×Q8—41no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—21no—
101C22×D4—22no—
102C23×C4—42no—
103C2×D8—28no—
104C2×D8—44no—
105C2×D8—48no—
106C2×D8—48no—
107C2×D8—44no—
108C2×D8—84no—
109C22×C8C2×D414yes—
110C22×C8C2×D414yes—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C4⋊Q8—22no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—44no—
117C4⋊Q8—84no—
118#127—22no—
119#127—22no—
120#127—22no—
121#127—21no—
122#127—22no—
123#127—21no—
124#127#4114yes—
125#124—44no—
126#124—44no—
127C4×D4—48no—
128C4×D4—48no—
129C4×D4—84no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—42no—
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2411yescharacteristic
151#106—44no—
152C4⋊C8—44no—
153#103—44no—
154#103—44no—
155#102—48no—
156#102—48no—
157#102—44no—
158#102—44no—
159#102—84no—
160#102—84no—
161#102—84no—
162#97—24no—
163#97—24no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—21no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—21no—
171C24—42no—
172C24—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—82no—
178C2×Q8—84no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—41no—
183C2×D4—42no—
184C2×D4—41no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—44no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—84no—
204C22×C4—84no—
205C22×C4—84no—
206C22×C4#12711yescharacteristic
207C22×C4#12711yescharacteristic
208C22×C4#12712yes—
209C22×C4C22×D411yescharacteristic
210C22×C4C22×D412yes—
211D8—416no—
212D8—816no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—24no—
221C2×C8—24no—
222C2×C8—24no—
223C2×C8—24no—
224C2×C8AGL(1,ℤ/8)14yes—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—82no—
235C4⋊C4—84no—
236C22⋊C4—84no—
237C22⋊C4—84no—
238C22⋊C4—82no—
239C22⋊C4—82no—
240C22⋊C4—84no—
241C22⋊C4—84no—
242C42—22no—
243C42—24no—
244C42—24no—
245C42AGL(1,ℤ/8)14yes—
246C42C2×#4112yes—
247C42H(5,𝔽2)11yescharacteristic
248C422−1+411yescharacteristic
249C23—44no—
250C23—82no—
251C23—82no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—81no—
257C23—81no—
258C23—82no—
259C23—82no—
260C23—81no—
261C23C2×#12711yescharacteristic
262Q8—82no—
263Q8—82no—
264Q8—164no—
265D4—48no—
266D4—48no—
267D4—48no—
268D4—82no—
269D4—84no—
270D4—84no—
271D4—84no—
272D4—82no—
273D4—84no—
274D4—84no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—168no—
292C2×C4#49614yes—
293C2×C4#52511yescharacteristic
294C2×C4#53711yescharacteristic
295C2×C4#54612yes—
296C2×C4#54812yes—
297C2×C4C2×AGL(1,ℤ/8)12yes—
298C2×C4C2×AGL(1,ℤ/8)12yes—
299C2×C4#57514yes—
300C2×C4#57614yes—
301C8—24no—
302C8—42no—
303C8—44no—
304C8—44no—
305C22—84no—
306C22—84no—
307C22—84no—
308C22—162no—
309C22—162no—
310C22—162no—
311C22—162no—
312C22—162no—
313C22—161no—
314C22—161no—
315C22—162no—
316C22—161no—
317C22#254011yescharacteristic
318C22C2×#49611yescharacteristic
319C22#315111yescharacteristic
320C22#315312yes—
321C22#315512yes—
322C4—24no—
323C4—41no—
324C4—41no—
325C4—161no—
326C4—161no—
327C4—162no—
328C4—161no—
329C4#310912yes—
330C4#325312yes—
331C4#331914yes—
332C4#337014yes—
333C2—164no—
334C2—321no—
335C2#1967312yes—
336C2#1972711yescharacteristic
337C2#1975011yescharacteristic
338C2#2838111yescharacteristic
339C2#2842212yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#10304C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #33230C2semidirect
2. #29139C2semidirect
3. #28615C2semidirect
4. #21690C2semidirect
5. #10686C2semidirect
6. #10677C2semidirect
7. #10307C2semidirect
9. #9838C2semidirect
10. #9832C2semidirect
11. #8263C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#332302
#291392
#286152
#216902
#106862
#106772
#103072
#103042
#98382
#98322
#82632